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In 1908 the Austrian artist Richard Gerstl committed suicide and changed musical history. It was a flamboyant death by anyone’s standards – after burning his paintings, he hanged himself naked in front of a full-length mirror. The cause of his emotional anguish was the end of his affair with the wife of the composer Arnold Schoenberg, Mathilde. When he heard the news, Schoenberg, already traumatised by his wife’s infidelity, contemplated suicide himself but, instead, sublimated his despair in a composition that took him for the first time into atonality.

Today it’s difficult to imagine how dramatic that first lurch was but, in classical music terms, it’s fair to say it was the equivalent of the splitting of the atom. Alex Ross, music critic of the New Yorker, summed up Schoenberg’s impact in his anecdote-stuffed, intellectually exhilarating romp through 20th-century classical music history, The Rest Is Noise: Listening to the 20th Century (2007). “Nothing in the annals of musical scandal,” he wrote, “from the first night of Stravinsky’s Rite of Spring to the release of the Sex Pistols’ Anarchy in the UK, rivals the ruckus that greeted Schoenberg early in his career.” At the premiere of the Second Quartet, the composition that marked Schoenberg’s departure from tonality, the critic Ludwig Karpath screamed out, “Stop it! Enough!” Five years later, at a concert featuring the work of Schoenberg and his pupils, the explosion of outrage was so great that the police had to be called in.

The paradox of atonality is that although it sounds like a casting into outer darkness (significantly, the image of the devil recurs frequently in Ross’s book), it’s also the point when rational mathematical calculations become more explicit in music. There has always been a connection between juggling numbers and juggling notes. Yet once composers such as Schoenberg, Webern, Berg and, later, Cage, Stockhausen and Boulez denounced traditional forms of rhythm and harmony, the search for what should replace them incorporated everything from coin-tossing to complex calculations with prime numbers. That’s why – as the Southbank Centre opens its year-long festival The Rest Is Noise inspired by Ross’s book – I find myself seeking a mathematician as a guide.

Not that Marcus du Sautoy – who, as well as being a professor of mathematics at Oxford university, has succeeded Richard Dawkins as Simonyi professor for the public understanding of science – can adequately be summed up by the word “mathematician”. For a start there’s the hurricane force of his personality (journalists have a lot of fun talking about his bright clothes), and the range of his ambition (theatre is a passion, as is the desire to be a chef). We meet at his home in Stamford Hill in north London and, as he waves his arms around talking about everything from turn-of-the-century polymath Poincaré to palindromes, his enthusiasm for music makes him resemble a follically challenged Simon Rattle.

I talk to du Sautoy about the riots that greeted Schoenberg’s work. Why did atonal music disturb people so much? “I think musicians wanted to upset audiences,” he replies. “They were breaking the complacency of previous music – people were expecting to reach one destination through harmony or rhythm, and suddenly they were pulled away to somewhere completely different. What’s so exciting for me is how different that soundscape was – it was a time of complete change. In some ways one does have to go through the whole gamut of history to understand 20th-century music.”

Luckily for wannabe contemporary music aficionados, the Southbank Centre is providing historical context through lectures, films and themed events: from Gertrude Stein to the imagery of the swastika, from the invention of the recording cylinder to racial segregation in America. The first section of the festival, Here Comes the 20th Century, looks at how the complacency du Sautoy describes – not just in music but in other aspects of western culture – was being dismantled. Talks will encompass the suffragette movement and the invention of radio. Ross himself lectures this weekend on the mood in Vienna as composers strove for a language that would distance them from their bourgeois predecessors. And du Sautoy is lecturing on Saturday on Einstein’s theory of special relativity, looking at how the resulting shifts in perception of time and space impacted on the work of composers such as Messiaen.

“This became a time when people were asking a lot of questions about our place in the universe,” he declares.

“Initially, I think that Henri Poincaré [the polymath whose book Science and Hypothesis inspired both Einstein and Picasso] had a bigger influence on the artistic world, because he was writing for them. I suspect that comes from Poincaré. With Messiaen, in Quartet for the End of Time, you have a sense of never-ending time that relates to Einstein but you have to be careful about saying how direct that relationship is.”

One of the recurring questions in The Rest Is Noise is whether there was a point when 20th-century classical music talked itself into an intellectual corner. The 1920s Berlin composer Kurt Weill compared musicians who, “filled with disdain for the public, work toward the solution of aesthetic problems as if behind closed doors” and those – like him – who “open up a connection with any kind of public”.

Although du Sautoy is charming and charismatic, there are points when he’s explaining the intricate maths of certain compositions where I feel I’m being given the crib card for an esoteric cult. But he is never less than entertaining to watch – at one point when he’s explaining symmetry and 12-tone technique he bends his left arm in so many directions I worry he’s going to dislocate his elbow.

I put it to him that Quartet for the End of Time, for instance, is about much more than physics and maths, not least because of the circumstances in which it was composed. As du Sautoy describes in his book The Number Mysteries, Messiaen wrote it while he was detained during the second world war in camp Stalag VIII-A in Silesia. He discovered he was imprisoned with fellow former conservatoire students, who included a violinist, a clarinettist and a cellist. A sympathetic German sergeant gave him the opportunity to compose – so that this stark, deeply original piece was first performed in front of prison guards and inmates, with Messiaen playing a “rickety upright piano”.

“Though there might be an academic angle, it should never be a primary experience for someone listening to it,” du Sautoy concurs. “In the piano part Messiaen has used an intellectual technique using prime numbers to create an effect. But he doesn’t want you to say ‘Aha, I get this music, I can hear the prime numbers 17 and 29.’ No, the point is you get this sense that things aren’t quite meshing. It’s about creating a sense of unease.”

He’s adamant, however, that music is a good way to tempt the curious into mathematics. “With modern classical music, the more work you put in the more you get out,” he declares. “For me mathematics is a very emotional, expressive medium. What’s interesting is that Messiaen was often drawn to mathematical structures not because he understood the maths but because of their universality. That’s why I respond to a composer like him. Because he’s playing the same games that I’m playing.”